Vertices of ideals of a \({\mathfrak p}\)-adic number field (Q1087585)

Let \(K/k\) be a finite normal extension with Galois group \(G\) over a \({\mathfrak p}\)-adic number field, and \(\pi\) be a prime element of the ring of integers in \(K\). The author intends to study the structure of ideals \((\pi^ i)\) of \(K\) as Galois module, and first defines ''the vertex \(V(\pi^ i)\) of an ideal \((\pi^ i)\)'' as the minimal normal subgroup \(S\) of \(G\) such that \((\pi^ i)\) is \((G,S)\)-projective, i.e. relatively projective with respect to a subgroup \(S\) of \(G\). He provides explicitly some fundamental relations between the vertex \(V(\pi^ i)\) and the j-th ramification group \(G_ j\) of \(K/k\). One of them is \(G_ 1\supseteq V(\pi^ i)\supseteq G_ 2\) for any i, which is a generalization of \textit{A. Fröhlich}'s result [cf. Algebr. Zahlentheorie, Ber. Tagung math. Forschinst. Oberwolfach 1964, 59-82 (1966; Zbl 0199.097)].